Thermal emission is the radiation of electromagnetic waves from hot objects. The promise of thermal-emission engineering for applications in energy harvesting, radiative cooling, and thermal camouflage has recently led to renewed research interest in this topic. However, accurate and precise measurements of thermal emission in a laboratory setting can be challenging in part due to the presence of background emission from the surrounding environment and the measurement instrument itself. This problem is especially acute for thermal emitters that have unconventional temperature dependence, operate at low temperatures, or are out of equilibrium. In this paper, we describe, recommend, and demonstrate general procedures for thermal-emission measurements that can accommodate such unconventional thermal emitters.